How to mirror cut or modification made to basic shape?
Blender wont mirror operations made on a mesh but will mirror the mesh itself, in the state it is when the mirror is applied.

So, most of time, you may anticipate that your shape will be symetrical and define the mirror appropriatly.

By default, mirror is applied along the x-axis (1) and at the center point of your mesh (the yellow dot you can see in object mode marked 2 here).
In this example the mirror position corresponds to the plane defined by X=3, as the mirror is on X and is defined relatively to the object's center at the position (3, -1, 0).



This previous image shows a cube. But in fact, only a half cube is defined in this example :



This half cube is not a plain figure, but some kind of open box as a face is missing (see above) because the final mirrored cube does not need an inner face (at least this is the most common situation but this is not mandatory).

Once the mirror is placed and if you want to use the boolean modifier, some tricky things may happen (boolean modifier is not allways so simple) : or the intersecting object is not itself mirrored or it is :



If it is not, all is fine, and you obtain this kind of result when you apply the boolean modifier :



But if it is the two shape will be intricated, and that's probably not what you want :



So to avoid that, an easier way is to work on a half of the model (not mirrored) and add the mirror modifier once finished with the boolean.

In my opinion (but the following remark may be considered as subjective), using the boolean modifier is not the simpliest way to start using Blender, even if it seems "simple" at first sight (it may also have some other unwanted consequences).
So for a begining, I'd suggest to keep using the mirror modifier (if you need it) but go in edit mode to model your shape manually.